Fat-Tailed Uncertainty in the Economics of Catastrophic Climate Change
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275
Symposium:
Fat Tails and the Economics of Climate Change
Fat-Tailed Uncertainty in the
Economics of Catastrophic Climate
Change
Martin L. Weitzman*
Introduction
I believe that the most striking feature of the economics of climate change is that its extreme
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downside is nonnegligible. Deep structural uncertainty about the unknown unknowns of
what might go very wrong is coupled with essentially unlimited downside liability on possible
planetary damages. This is a recipe for producing what are called ‘‘fat tails’’ in the extremes of
critical probability distributions. There is a race being run in the extreme tail between how
rapidly probabilities are declining and how rapidly damages are increasing. Who wins this
race, and by how much, depends on how fat (with probability mass) the extreme tails are. It is
difficult to judge how fat the tail of catastrophic climate change might be because it represents
events that are very far outside the realm of ordinary experience.
In this article, which is part of a symposium on Fat Tails and the Economics of Climate
Change, I address some criticisms that have been leveled at previous work of mine on fat
tails and the so-called ‘‘dismal theorem.’’1 At first, I was inclined to debate some of the critics
and their criticisms more directly. But, on second thought, I found myself anxious not to be
drawn into being too defensive and having the main focus be on technical details. Instead, I
am more keen here to emphasize anew and in fresh language the substantive concepts that, I
think, may be more obscured than enlightened by a debate centered on technicalities. I am
far more committed to the simple basic ideas that underlie my approach to fat-tailed un-
certainty and the economics of catastrophic climate change than I am to the particular
*Department of Economics, Harvard University; e-mail: mweitzman@harvard.edu
Without blaming them for the remaining deficiencies in this article, I am extremely grateful for the con-
structive comments on an earlier version by James Annan, Daniel Cole, Stephen DeCanio, Baruch Fischoff,
Don Fullerton, John Harte, William Hogan, Matthew Kahn, David Kelly, Michael Oppenheimer, Robert
Pindyck, Joseph Romm, and Richard Tol.
1
This symposium also includes articles by Nordhaus (2011) and Pindyck (2011). The ‘‘dismal theorem,’’
introduced in Weitzman (2009a), will be discussed later in this article.
Review of Environmental Economics and Policy, volume 5, issue 2, summer 2011, pp. 275–292
doi:10.1093/reep/rer006
Ó The Author 2011. Published by Oxford University Press on behalf of the Association of Environmental and Resource
Economists. All rights reserved. For permissions, please email: journals.permissions@oup.com276 M. L. Weitzman
mathematical form in which I have chosen to express them. These core concepts could have
been wrapped in a variety of alternative mathematical shells—and the particular one that I
chose to use previously is somewhat arbitrary. The implications are roughly similar, irre-
spective of formalization. Some technical details are unavoidable, but if I can give the un-
derlying concepts greater intuitive plausibility, then I believe that this set of ideas will
become more self-evident and more self-evidently resistant to several of the criticisms that
have been leveled against it.
In the next section, I present an intuitive–empirical argument that deep structural un-
certainty lies at the heart of climate change economics. Then, in the following section, I try
to explain some of the theory behind fat-tailed extreme events and discuss some possible
implications for the analysis of climate change. I offer a few summary remarks in the final
section.
Some Empirical Examples of Deep Structural Uncertainty
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about Climate Extremes
In this section I try to make a heuristic–empirical case (in the form of five ‘‘stylized facts’’) for
there being big structural uncertainties in the economics of extreme climate change. I will argue
on intuitive grounds that the way in which this deep uncertainty is conceptualized and formal-
ized should influence the outcomes of any reasonable benefit–cost analysis (BCA) of climate
change. Furthermore, I will argue that the seeming immunity of the ‘‘standard’’ BCA to the
possibility of extreme outcomes is both peculiar and disturbing. My arguments in this section
are not intended to be airtight or rigorous. Rather, this is an intuitive presentation based on some
very rough stylized facts.
By BCA of climate change, I mean, in the widest sense, some overall economic analysis
centered on maximizing (or at least comparing) welfare. My notion of BCA in the present
context is so broad that it overlaps with an integrated assessment model (IAM), and here I
treat the two as essentially interchangeable. I begin by setting up a straw man that I will label
the ‘‘standard BCA of climate change.’’ Of course, there is no ‘‘standard BCA of climate
change,’’ but I think this is an allowable simplification for purposes of exposition here.
We all know that computer-driven simulations are dependent upon the core assumptions
of the underlying model. The intuitive examples presented below are frankly aimed at sowing
a few seeds of doubt that the ‘‘standard BCA of climate change’’ fairly represents structural
uncertainties about extreme events, and that therefore its conclusions might be less robust
than is commonly acknowledged. I argue not that the standard model is wrong or even
implausible, but rather that it may not be robust with respect to the modeling of catastrophic
outcomes. I will try to make my case by citing five aspects of the climate science and economics
that do not seem to me to be adequately captured by the standard BCA. The five examples—
which I call Exhibits 1, 2, 3, 4 and 5—are limited to structural uncertainty concerning the
modeling of climate disasters. While other important aspects of structural uncertainty might
also be cited, I restrict my stylized facts to these five examples. In the spirit of performing
a kind of ‘‘stress test’’ on the standard BCA, I naturally concentrate on things that might go
wrong rather than things that might go right.Fat-Tailed Uncertainty in the Economics of Catastrophic Climate Change 277
Exhibit 1: Past and Present Greenhouse Gas Concentrations
Exhibit 1 concerns the atmospheric level of greenhouse gases (GHGs) over time. Ice-core
drilling in Antarctica began in the late 1970s and is still ongoing. The record of carbon dioxide
(CO2) and methane (CH4) trapped in tiny ice-core bubbles currently spans 800,000 years.2
It is important to recognize that the numbers in this unparalleled 800,000-year record of
GHG levels are among the very best data that exist in the science of paleoclimate. Almost
all other data (including past temperatures) are inferred indirectly from proxy variables,
whereas these ice-core GHG data are directly observed.
The preindustrial revolution level of atmospheric CO2 (about two centuries ago) was 280
parts per million (ppm). The ice-core data show that CO2 varied gradually during the last
800,000 years within a relatively narrow range roughly between 180 and 280 ppm and has
never been above 300 ppm. Currently, CO2 is over 390 ppm and climbing steeply. In 800,000
years, methane has never been higher than 750 parts per billion (ppb), but now this extremely
potent GHG, which is twenty-two times more powerful than CO2 (per century), is at about
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1,800 ppb. The sum total of all carbon dioxide equivalent (CO2e) GHGs is currently at about
450 ppm. An even more startling contrast with the 800,000-year record is the rate of change of
GHGs: increases in CO2 were below (and typically well below) 25 ppm within any past
subperiod of 1,000 years, while now CO2 has risen by over 25 ppm in just the last ten years.
Thus, anthropogenic activity has very rapidly elevated atmospheric CO2 and CH4 to levels
very far outside their natural range. The unprecedented scale and speed of GHG increases
brings us into uncharted territory and makes predictions of future climate change
very uncertain. Looking ahead a century or two, the levels of atmospheric GHGs that
may ultimately be attained (unless decisive measures are undertaken) have likely not existed
for tens of millions of years, and the speed of this change may be unique on a time scale of
hundreds of millions of years.
Remarkably, the ‘‘standard BCA of climate change’’ takes little account of the magnitude
of the uncertainties involved in extrapolating future climate change so far beyond past
experience. Perhaps even more surprising, the gradual tightening of GHG emissions, which
emerges as optimal policy from the ‘‘standard’’ BCA, typically attains stabilization at levels of
CO2 that approach 700 ppm (and levels of CO2e that are even higher). The ‘‘standard’’ BCA
thus recommends subjecting the Earth’s system to the unprecedented shock of instanta-
neously (in geological terms) jolting atmospheric stocks of GHGs up to two-and-a-half times
above their highest level over the past 800,000 years—without mentioning the unprecedented
nature of this unique planetary experiment. This is my Exhibit 1.
Exhibit 2: The Uncertainty of the Climate Change Response
Exhibit 2 concerns the highly uncertain climate change response to the kind of unprecedented
increases in GHGs that were described in Exhibit 1. For specificity, I focus on the uncertainty
of so-called ‘‘equilibrium climate sensitivity,’’ which is a key macro-indicator of the eventual
temperature response to GHG changes. This is a good example of a ‘‘known unknown.’’
2
My numbers are taken from Dieter et al. (2008) and supplemented by data from the Keeling curve for more
recent times (available online at: ftp://ftp.cmdl.noaa.gov/ccg/co2/trends/co2_mm_mlo.txt).278 M. L. Weitzman
However, it should be understood that under the rubric of climate sensitivity, I am trying to
lump together an entire suite of other uncertainties, including some nonnegligible unknown
unknowns. The insights and results of this Exhibit 2 are not intended to stand or fall on the
single narrow issue of accurately modeling the effects of uncertain climate sensitivity. Rather,
equilibrium climate sensitivity is to be understood here as a prototype example of uncertainty,
or a metaphor, which is being used to illustrate much more generic issues concerning the
economics of uncertain climate change.
The Intergovernmental Panel on Climate Change (IPCC-AR4 2007) defines equilibrium cli-
mate sensitivity (hereafter S) this way: ‘‘The equilibrium climate sensitivity is a measure of the
climate system response to sustained radiative forcing. It is not a projection but is defined as
the global average surface warming following a doubling of carbon dioxide concentrations. It
is likely to be in the range 2–4.5°C with a best estimate of 3°C, and is very unlikely to be less than
1.5°C. Values substantially higher than 4.5°C cannot be excluded, but agreement of models with
observations is not as good for those values.’’ The actual empirical reason for why the upper tails of
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these probability distributions are long and seem heavy with probability dovetails with Bayesian
theory: inductive knowledge is always useful, of course, but simultaneously it is limited in what it
can tell us about extreme events outside the range of experience. In such situations, one is forced
back onto depending more than one might wish upon prior beliefs in the form of a prior prob-
ability distribution, which, of necessity, is largely subjective and relatively diffuse.
Thus, any curve-fitting exercise attempting to attribute probabilities to S 4.5°C, such as
I am doing here, is little more than conjectural speculation. My purpose is merely to show that
critical results can depend on seemingly casual decisions about how to model tail probabilities. To
illustrate some striking implications for the analysis of climate change, I contrast the use of two
familiar probability distributions to represent the upper-half tail of climate sensitivity above the
median:(a) the Pareto(or Poweror Polynomial) distribution,subscripted P, which is the prototype
exampleofafatuppertail;and(b)theNormaldistribution,subscriptedN,whichistheprototypeof
a thin upper tail. (By common definition, probabilities that decline exponentially or faster, like the
Normal distribution, arethin tailed, while probabilitiesthat declinepolynomiallyor slower, likethe
Pareto distribution, are fat tailed). The IPCC defines ‘‘likely’’ as a probability above 66 percent but
below 90 percent, which would mean that the probability that climate sensitivity is greater than
4.5°C (Prob[S 4.5°C]) is between 5 percent and 17 percent. A more recent average estimate of
fourteen leading climate scientists is Prob[S 4.5°C] ¼ 23 percent (Zickfeld et al. 2010). Here I
choose Prob[S 4.5° C] ¼ 15 percent and I calibrate parameters of both probability distributions
so that Prob[S 3°C] ¼ 0.5, and Prob[S 4.5°C] ¼ 0.15.3
Table 1 presents cumulative probabilities of climate sensitivity for the Pareto and Normal
distributions. Table 1 indicates a tremendous difference in upper tail behavior between the
fat-tailed Pareto distribution and the thin-tailed Normal distribution. I think that the Pareto
distribution of climate sensitivity has a disturbingly large amount of probability (ProbP) in its
upper tail. There is no consensus on how to aggregate the results of many different climate
3
I lean more toward Prob[S 4.5°C] 17% than toward Prob[S 4.5°C] 5% because, for a time horizon
of a century and a half or so, it is plausible that the more inclusive and larger ‘‘earth system sensitivity’’ (which
includes slow feedbacks like methane releases) matters more than the ‘‘fast-feedback equilibrium sensitivity’’
that IPCC-AR4 refers to. For more on the distinction between fast- and slow-feedback climate sensitivities,
see, for example, Hansen et al. (2008).Fat-Tailed Uncertainty in the Economics of Catastrophic Climate Change 279
Table 1 Prob½S S^ for fat-tailed Pareto and thin-tailed Normal distributions
S^ ¼ 3°C 4.5°C 6°C 8°C 10°C 12°C
^
ProbP ½S S 0.5 0.15 0.06 0.027 0.014 0.008
ProbN ½S S^ 0.5 0.15 0.02 0.003 7 107 3 1010
sensitivity studies into one overarching probability density function (PDF), and there is much
controversy about how it might be done. But for what it is worth (perhaps very little), the
median upper 5 percent probability level over all 22 climate sensitivity PDFs cited in IPCC-
AR4 is 6.4°C, which fits with the Pareto PDF in Table 1 above.4
Table 2 presents some values of probabilities of eventual increased global mean surface
temperatures (T) as a function of stationary greenhouse gas concentrations (G).5 The first
row of Table 2 represents levels of G (measured in ppm of CO2e). The second row indicates
the median equilibrium temperature as a function of stabilized GHG stocks. The remaining
rows indicate the probabilities (ProbP and ProbN) of achieving at least the steady-state
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temperature increase represented by the entries in the table (5°C and 10°C) for both of
the chosen PDFs (Pareto ¼ P ¼ fat tail; Normal ¼ N ¼ thin tail).
What is especially striking to me about Table 2 is the reactiveness of high-temperature
probabilities to the level of GHGs. The fat-tailed Pareto case seems especially worrisome.
One implication is that an optimal policy might be expected to keep GHG levels down
and be much less casual about letting CO2e levels exceed 700 ppm than the ‘‘standard’’
BCA. I believe that Table 2 could be taken as some indirect evidence that the main purpose
of keeping GHG concentrations down is effectively to buy insurance against catastrophic
global warming. The above examples of the highly uncertain eventual temperature response
to unprecedented increases in GHGs constitute my Exhibit 2.
Exhibit 3: A Physical Basis for Catastrophic Outcomes
Exhibit 3 concerns possibly disastrous releases over the long run of bad feedback components of
the carbon cycle that are currently omitted from most general circulation models. The chief
concern here is that there may be a significant supplementary component that conceptually
Table 2 Probabilities of exceeding T ¼ 5°C and T ¼ 10°C for given G ¼ ppm of CO2e
G: 400 500 600 700 800 900
Median T 1.5° 2.5° 3.3° 4.0° 4.5° 5.1°
ProbP[T 5°C] 1.5% 6.5% 15% 25% 38% 52%
ProbN[T 5°C] 106 2.0% 14% 29% 42% 51%
ProbP[T 10°C] 0.20% 0.83% 1.9% 3.2% 4.8% 6.6%
ProbN[T 10°C] 1030 1010 105 0.1% 0.64% 2.1%
4
For details, see Weitzman (2009a).
5
Table 1 indicates probabilities for climate sensitivity, which correspond to GHG levels of 560 ppm of CO2e
(i.e., a doubling of the preindustrial revolution level of 280 ppm). The different values of GHG concentrations
in Table 2 reflect the correspondingly different probabilities of temperatures responses, in proportion to the
logarithm of GHG concentrations relative to 280 ppm.280 M. L. Weitzman
should be added on to the so-called ‘‘fast feedback’’ equilibrium concept that IPCC-AR4 works
with. This omitted component (which would be part of a more inclusive slow-feedback
generalization called ‘‘earth system sensitivity’’) includes the powerful self-amplification
potential of greenhouse warming that is due to heat-induced releases of sequestered carbon.
One vivid example is the huge volume of GHGs currently trapped in tundra permafrost and
other boggy soils (mostly as methane, a particularly potent GHG). A more remote (but even
more vivid) possibility, which in principle should also be included, is heat-induced releases of
the even vaster offshore deposits of methane trapped in the form of clathrates.6
There is a very small and unknown (but decidedly nonzero) probability over the long run of
having destabilized methane from these offshore clathrate deposits seep into the atmosphere
if the temperature of the waters bathing the continental shelves increases just slightly. The
amount of methane involved is huge, although it is not precisely known. Most estimates place
the carbon-equivalent content of methane hydrate deposits at about the same order of mag-
nitude as all other fossil fuels combined. Over the long run, a methane outgassing–amplifier
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process could potentially precipitate a disastrous strong positive feedback warming. If it
occurred at all, such an event would likely take centuries to materialize because the presumed
initiator would be the slow-acting gradual warming of ocean waters at the depths of the
continental shelves. Thus, while it is a low-probability event that might only transpire
centuries from now (if at all), the possibility of a climate meltdown is not just the outcome
of a mathematical theory but has a real physical basis. Other examples of an actual physical
basis for catastrophic outcomes could be cited, but this one will do here. This is my Exhibit 3.
Exhibit 4: Damages of Extreme Climate Change
Exhibit 4 concerns what I view as a somewhat cavalier treatment in the literature of damages
or disutilities from extreme temperature changes. The ‘‘standard’’ BCA damages function
reduces welfare-equivalent output at mean global temperature change T by a quadratic-
polynomial multiplier of the form M(T) ¼ aT2/(1 þ aT2).7 The results in terms of fractional
damages to output are indicated by M(T) in Table 3.
I do not find the numbers in Table 3 convincing for higher temperatures. At an extraordi-
narily high global average temperature change of T ¼ 10°C, the welfare-equivalent damage in
Table 3 is a loss of ‘‘only’’ 19 percent of world output at the time of impact. If the annual growth
Table 3 Multiplicative-quadratic damages M(T) (as fraction of output)
T 2°C 4°C 6°C 8°C 10°C 12°C
M(T) 1% 4% 8% 13% 19% 26%
6
Clathrates (or hydrates) are methane molecules that are boxed into a semistable state by being surrounded
by water molecules under high pressure and low temperatures. For more about methane clathrates, see
Archer (2007) and the recent article by Shakhova et al. (2010).
7
For the sake of concreteness, I take the value of the parameter a used in the latest version of Nordhaus’s well-
known DICE model (Nordhaus 2008). The DICE model is perhaps the most famous IAM in the economics
of climate change. The value a ¼ 0.002388 was used to generate his figure 3-3 on page 51. I hasten to add in
fairness that this specification was intended only to capture low temperature damages and was never intended
to be extrapolated to very high temperature changes.Fat-Tailed Uncertainty in the Economics of Catastrophic Climate Change 281
rate is, say, 2 percent and the time of impact is, say, two centuries from now, then the welfare
difference between no temperature change in two hundred years and a temperature change of
10°C in two hundred years is the difference between output that is fifty-five times larger than
current levels and output that is forty-four times larger than current levels. This is equivalent
to a reduction in projected annual growth rates from 2 percent to 1.9 percent. Such a mild kind
of damages function is practically preordained to make extreme climate change look empirically
negligible, almost no matter what else is assumed. Conversely, it turns out that fat-tailed temper-
ature PDFs are not by themselves sufficient to make extreme climate change have empirical ‘‘bite’’
without a damages function that is far more immiserizing at high-temperature changes.
So what should the damages function be for very high temperatures? No one knows, of
course. Taking an extreme example, suppose for the sake of argument that average global tem-
peratures were to increase by the extraordinary amount of 10°C (with a low probability, and
occurring centuries hence). While it is true that people live very well in places where the mean
temperature is 10°C higher than in Yakutsk, Siberia, I do not think that this type of analogy
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justifies using a comparative geography approach for estimating welfare equivalent damages
from an average global temperature change of 10°C. Global mean temperatures involve a double
averaging: across space and over time. A ‘‘damages function’’ is a reduced form representing
global welfare losses from global average temperatures, which subsumes a staggering amount
of regional, seasonal, and even daily weather heterogeneity. Regional and seasonal climate
changes are presumably much more unpredictable than global average surface temperatures.
There is just too much structural uncertainty and too much heterogeneity to put trustworthy
bounds on the unprecedented, almost unimaginable, changes to planetary welfare that would
result from average global temperatures increasing by 10°C. When there is such great uncer-
tainty about catastrophic damages, and when the damages function for high-temperature
changes is so conjectural, the relevant degree of risk aversion, yet another important unknown
here, will tend to play a significant role in an economic analysis of climate change.
Of course, I have no objective way to determine the magnitudes of high-temperature damages.
The last time the world witnessed periods where global average temperatures were very roughly
10°C or so above the present was during the Eocene epoch 55–34 million years ago. During
these warming periods, the earthwas ice free, while palm trees and alligators lived near the North
Pole. The Eocene was also the last epoch in which there were geologically rapid increases in mean
global temperatures of magnitude 5°C or so above an already warm background. Such hyper-
thermal events occurred over an average period of very roughly 100,000 years or so, which is
extremely gradual compared to current worst-case anthropogenically induced trajectories. It is
unknown what exactly triggered these Eocene temperature increases, but they were accompa-
nied by equally striking atmospheric carbon increases. One likely culprit is the strong feedback
release of large amounts of methane hydrates from clathrate deposits (Exhibit 3), which is
a nonnegligible possibility over the next century or two if current GHG emissions trends
are extrapolated. The major point here is that relatively rapid changes in global average temper-
atures of 5°C above present values are extremely rare events and are extraordinarily far outside
the scope of human experience. For huge temperature increases such as T 10°C, the planetary
effects are even more difficult to imagine. To find a geologically instantaneous increase in average
global temperatures of magnitude T 10°C, one would have to go back hundreds of millions of
years.282 M. L. Weitzman
For me, 10°C offers both a vivid image and a reference point, especially in light of a recent
study, which estimated that global average temperature increases of 11–12°C (with, impor-
tantly, accompanying humidity in the form of a high wet-bulb temperature8) would exceed
an absolute thermodynamic limit to metabolic heat dissipation (Sherwood and Huber 2010).
Beyond this threshold, represented by a wet-bulb temperature of 35°C, more than half of to-
day’s human population would be living in places where, at least once a year, there would be
periods when death from heat stress would ensue after about six hours of exposure. (By
contrast, the highest wet-bulb temperature anywhere on Earth today is about 30°C).
Sherwood and Huber (2010) further emphasize: ‘‘This likely overestimates what could
practically be tolerated: Our [absolute thermodynamic] limit applies to a person out of
the sun, in a gale-force wind, doused with water, wearing no clothing and not working.’’ Even
at wet-bulb temperatures, much lower than 35°C, human life would become debilitating and
physical labor would be unthinkable. The massive unrest and uncontainable pressures this
might bring to bear on the world’s human population are almost unimaginable. The Earth’s
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ecology, whose valuation is another big uncertainty, would be upended. Thus, a temperature
change of 10°C would appear to represent an extreme threat to human civilization and
global ecology as we now know it, even if it might not necessarily mean the end of Homo
sapiens as a species.
It must be emphasized strongly that very high atmospheric temperature changes such as
T ¼ 10°C would likely take several centuries to attain. The higher the limiting temperature,
the longer it takes to achieve equilibrium because the oceans will first have to absorb the
enormous amounts of heat being generated. Alas, if the oceans are building up enormous
amounts of heat it could set in motion irreversible long-term methane clathrate releases from
the continental shelves along with some other nasty surprises. Thus, overall damages gen-
erated by equilibrium T ¼ 10°C are best conceptualized as associated with being on the tra-
jectory whose asymptotic limiting atmospheric temperature change is T ¼ 10°C.
As noted above, the ‘‘standard’’ BCA damages function reduces welfare-equivalent
consumption by a quadratic-polynomial multiplier. This essentially describes a single-
attribute utility function, or, equivalently, a multiattribute utility function with strong
substitutability between the two attributes of consumption and temperature change. This
would be an appropriate formulation if the main impact of climate change is, say, to drive
up the price of food and increase the demand for air conditioning. This particular choice of
functional form allows the economy to easily substitute higher consumption for higher
temperatures, since the limiting elasticity of substitution between consumption and higher
temperatures is one (due to the multiplicative-polynomial assumption). However, very
different optimal policies can be produced when other functional forms are used to express
the disutility of disastrously high temperatures. For example, suppose that instead of being
multiplicatively separable (as in the ‘‘standard’’ BCA), the disutility of temperature change is
instead additively separable. This means that welfare is the analogous additively separable
arithmetic difference between a utility function of consumption and a quadratic loss func-
tion of temperature changes. This specification amounts to postulating a genuine
8
Wet-bulb temperature is essentially measured by a thermometer whose bulb is encased in a wet sock being
cooled by a very strong wind. Above a wet-bulb temperature of 35°C, humans cannot shed enough core body
heat to live—even under ideal circumstances.Fat-Tailed Uncertainty in the Economics of Catastrophic Climate Change 283
multiattribute utility function that describes a situation where the main impact of climate
change is on things that are not readily substitutable with material wealth, such as bio-
diversity and health. If the utility function has a constant coefficient of relative risk aversion
of two, it implies an elasticity of substitution between consumption and temperature
change of one half. Empirically, using this additive form—even without any uncertainty
concerning temperatures—prescribes a significantly more stringent curtailment of GHG
emissions than what emerges from the analogous multiplicative form of the ‘‘standard’’
BCA.9 Allowing for the possibility of high temperatures (even with low probabilities)
would presumably exaggerate this difference between the additive and multiplicative func-
tional forms yet more. Such fragility to basic functional forms is disturbing because we
cannot know with confidence which specification of the utility function is more
appropriate.
The above discussions and examples of nonrobustness with respect to a damages function
for high temperatures constitutes my Exhibit 4.
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Exhibit 5: Discounting the Distant Future
Exhibit 5 concerns the notorious issue of how to discount the distant future. The effects of
global warming and climate change will be spread out over centuries and even millennia from
now. The logic of compounding a constant positive interest rate forces us to say that what one
might conceptualize as monumental—even earth-shaking—events, such as disastrous cli-
mate change, do not much matter when they occur in the deep future. Perhaps even more
disturbing, when exponential discounting is extended over very long time periods there is
a truly extraordinary dependence of BCA on the choice of a discount rate. Seemingly insig-
nificant differences in discount rates can make an enormous difference in the present dis-
counted value of distant future payoffs. In many long-run situations, almost any answer to
a BCA question can be defended by one or another particular choice of a discount rate. This is
true in general, but it is an especially acute problem when distant future events like climate
change (especially catastrophic climate change) are being discounted.
There is a high degree of uncertainty about what should be taken as the appropriate real rate
of return on capital in the long run, accompanied by much controversy about its implications
for long-run discounting. There is no deep reason or principle that allows us to extrapolate
past rates of return on capital into the distant future. The industrial revolution itself began
some two centuries ago in Britain and only slowly thereafter permeated throughout the
world. The seeming trendlessness of some past rates of return is a purely empirical reduced
form observation, which is not based on any underlying theory that would allow us to
9
With a coefficient of relative risk aversion of two, the above additively separable specification is mathemat-
ically equivalent to the constant elasticity of substitution (CES) specification of Sterner and Persson (2008)
with CES one half. In their pioneering study, Sterner and Persson showed empirically—by plugging it into
NordhausÕs deterministic DICE model—that their CES (or, equivalently, my additive) welfare specification
prescribes a significantly more aggressive policy response to global warming (with a significantly higher car-
bon tax) than the analogous multiplicative specification of the ‘‘standard’’ BCA. The disturbingly significant
distinction between utility functions that are multiplicatively separable in consumption and temperature
change and those that are additively separable in consumption and temperature change is explored in some
detail in Weitzman (2009b).284 M. L. Weitzman
confidently project past numbers far into the future. There are a great many fundamental
factors that cannot easily be extrapolated, just one of which is the unknown future rate
of technological progress. Even leaving aside the question of how to project future interest
rates, additional issues for climate change involve which interest rate to choose out of
a multitude of different rates of return that exist in the real world.10 Furthermore, there
is a strong normative element having to do with what is the ‘‘right’’ rate, which brings an
ethical dimension to discounting climate change across many future generations that is
difficult to evaluate and incorporate into standard BCA. This normative issue is further
complicated when the event affecting future generations is a low-probability, high-impact
catastrophic outcome. If the utility function is additively separable in damages and consump-
tion, then the appropriate interest rate for discounting damages is not the rate of return on
capital but rather the rate of pure time preference, which for a normative evaluation of climate
change is arguably near zero.
The constant interest rates used for discounting in the ‘‘standard’’ BCA would be viewed by
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many people as severely biasing BCA toward minimizing into near nothingness the present
discounted value of distant future events, such as climate change. This kind of exponential
discounting, perhaps more than anything else, makes scientists and the general public
suspicious of the economist’s ‘‘standard’’ BCA of climate change, since it trivializes even truly
enormous distant future impacts. Honestly, I think that there are few economists who do not
feel uneasy about evaluating distant future climate change impacts this way. One line of
research, in which I have been involved, shows that when the discount rate itself is uncertain,
it implies that the ‘‘effective’’ discount rate declines over time to its lowest possible value (see
Weitzman 1998). Empirically, this effect can be quite powerful (see Weitzman 2010). The
driving force is a ‘‘fear factor’’ that derives from risk aversion to permanent productivity
shocks representing bad future states of the world. Whatever its source, the unknown
discount rate (and the extraordinary sensitivity of policy to its choice) is yet another big struc-
tural uncertainty in the economic analysis of climate change, especially when evaluating
possible catastrophes. This is my Exhibit 5.
A Long Chain of Structural Uncertainties
The above five exhibits could readily be extended to incorporate yet more examples of struc-
tural uncertainty, but enough is enough. To summarize, the economics of climate change
consists of a very long chain of tenuous inferences fraught with big uncertainties in every
link: beginning with unknown base-case GHG emissions; compounded by big uncertainties
about how available policies and policy levers will affect actual GHG emissions; compounded
by big uncertainties about how GHG flow emissions accumulate via the carbon cycle into
GHG stock concentrations; compounded by big uncertainties about how and when GHG
stock concentrations translate into global average temperature changes; compounded by
big uncertainties about how global average temperature changes decompose into specific
changes in regional weather patterns; compounded by big uncertainties about how adapta-
tions to, and mitigations of, climate change damages at a regional level are translated into
regional utility changes via an appropriate ‘‘damages function’’; compounded by big
10
For more on this, see Weitzman (2007).Fat-Tailed Uncertainty in the Economics of Catastrophic Climate Change 285
uncertainties about how future regional utility changes are aggregated into a worldwide utility
function and what its overall degree of risk aversion should be; compounded by big uncer-
tainties about what discount rate should be used to convert everything into expected present
discounted values. The result of this lengthy cascading of big uncertainties is a reduced form
of truly extraordinary uncertainty about the aggregate welfare impacts of catastrophic climate
change, which is represented mathematically by a PDF that is spread out and heavy with
probability in the tails.
What I would wish the reader to take away from these five exhibits is the notion that the
seeming immunity of the ‘‘standard’’ BCA to such stylized facts is both peculiar and disturb-
ing. An unprecedented and uncontrolled experiment is being performed by subjecting planet
Earth to the shock of a geologically instantaneous injection of massive amounts of GHGs. Yet,
the standard BCA seems almost impervious to the extraordinarily uncertain probabilities and
consequences of catastrophic climate change. In light of the above five exhibits of structural
uncertainty, the reader should feel intuitively that it goes against common sense that a climate
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change BCA does not much depend upon how potential future disasters are modeled and
incorporated into the BCA. This uneasy intuitive feeling based on stylized empirical facts
is my opening argument. I turn next to the theory.
The Dismal Theorem, Infinity, and BCA: A Theoretical
Framework
I begin this section by asking why it is relevant in the first place to have any supporting theory
if the five stylized fact exhibits from the previous section are convincing. Why aren’t these
stylized facts alone sufficient evidence that there is a problem with the ‘‘standard BCA of
climate change’’? My answer is that a combined theoretical plus empirical–intuitive argument
delivers a particularly powerful one–two punch at the treatment of structural uncertainty in
the standard BCA. In this respect, I believe that the whole of my argument is bigger than the
sum of its two parts. The theoretical part reinforces the empirical part by placing it within
a formal mathematical framework. When the intuitive exhibits are seen as reflecting some
formalized theoretical structure, then it becomes less easy to brush them aside as mere sniping
at an established model. In this theoretical section, as in the empirical section above, I
emphasize the intuitive plausibility of the case I am trying to make—focusing here on
the underlying logic that drives the theory.
In the previous section I argued that it is only common sense that climate change policy
implications should depend on the treatment of low-probability, extreme-impact outcomes.
The main question I attempted to address in Weitzman (2009a) was whether such intuitive
dependence is reflecting some deeper principle. My answer was that there is indeed a basic
underlying theoretical principle (the ‘‘dismal theorem,’’ hereafter DT) that points in this di-
rection. The simple logic can be grasped intuitively without using (or understanding) the
fancy math required to state and prove a formal version of the principle. In this section I
restate the theoretical arguments underlying the DT in what is hopefully a more intuitive
form.
Let welfare W stand for expected present discounted utility, whose theoretical upper bound
is B. Let D [ B W be expected present discounted dis utility. Here D stands for what might286 M. L. Weitzman
be called the ‘‘diswelfare’’ of climate change. Assume for the sake of argument that D is
‘‘essentially’’ unbounded in the particular case of climate change because global liability
is ‘‘essentially’’ unlimited in a worst-case scenario. (More later on what happens when
D is, technically, bounded.) Because the integral over a nonnegative probability measure
is one, the PDF of lnD must decline to zero. In other words, extreme outcomes can happen,
but their likelihood diminishes to zero as a function of how extreme the outcome might be.
The idea that extreme outcomes cannot be eliminated altogether but are hypothetically
possible with some positive probability is not at all unique to climate change. Almost nothing
in our world has a probability of exactly zero or exactly one. What is worrisome is not the fact
that the upper tail of the PDF of lnD is long (reflecting the fact that a meaningful bound on
diswelfare does not exist), but that it might be fat (reflecting the fact that the probability of
a catastrophic outcome is not sufficiently small to give comfort). The critical question, which
tail fatness quantifies, is how fast does the probability of a catastrophe decline relative to the
welfare impact of the catastrophe.
Unless otherwise noted, my default meaning of the term ‘‘fat tail’’ (or ‘‘thin tail’’)11 hence-
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forth concerns the upper tail of the PDF of lnD, resulting from whatever combination of
probabilistic temperature changes, temperature-sensitive damages, utility functions, dis-
counting, and so forth, makes this come about. This is the PDF that ultimately matters.
Empirically, it is not the fatness of the tail of the climate sensitivity PDF alone or the reactivity
of the damages function to high temperatures alone, or the degree of relative risk aversion
alone, or the rate of pure time preference alone, or any other factor alone, that counts, but
rather the interaction of all such factors in determining the upper-tail fatness of the PDF of
lnD. For example, other things being equal, the PDF of lnD will have a relatively fatter tail the
larger is the probability of high temperatures and the greater is the reactivity of the damages
function to high temperatures, but neither condition alone implies a fat-tailed PDF of lnD. It
may seem arcane, but the upper-tail fatness of the reduced form PDF of lnD is the core issue in
the economics of catastrophic climate change. Of course, it is extremely difficult to know the
fatness of the upper tail of the PDF of lnD, which is precisely the main point of this line of
research and this article.
In Weitzman (2009a), I indicated a theoretical tendency for the PDF of lnD to have a fat tail.
Conceptually, the underlying mechanism is fairly straightforward. Structural uncertainty
essentially means that the probabilities are unsure. A formal Bayesian translation might
be that the structural parameters of the relevant PDFs are themselves uncertain and have
their own PDFs. Weitzman (2009a) expressed this idea in a formal argument that the reduced
form ‘‘posterior predictive’’ PDF (in Bayesian jargon) of lnD tends to be fat tailed because the
structural parameters are unknown. Loosely speaking, the driving mechanism is that the
operation of taking ‘‘expectations of expectations’’ or ‘‘probability distributions of probability
11
There is some wiggle room in the definition of what constitutes a fat-tailed PDF or a thin-tailed PDF, but
everyone agrees that probabilities declining exponentially or faster (like the Normal) are thin tailed, while
probabilities declining polynomially or slower (like the Pareto) are fat tailed. The standard example of a fat-
tailed PDF is the power law (aka Pareto aka inverted polynomial) distribution, although, for example, a Stu-
dent-t or inverted-gamma PDF is also fat tailed. A normal or a gamma are examples of thin-tailed PDFs, as is
any PDF having finite supports, like a uniform distribution or a discrete-point finite distribution. Although
both PDFs must approach a limit of zero, the ratio of a fat-tailed probability divided by a thin-tailed prob-
ability goes to infinity in the limit.Fat-Tailed Uncertainty in the Economics of Catastrophic Climate Change 287
distributions’’ tends to spread apart and fatten the tails of the compounded posterior pre-
dictive PDF. From past samples alone, it is inherently difficult to learn enough about the
probabilities of extreme future events to thin down the ‘‘bad’’ tail of the PDF because we
do not have much data about analogous past extreme events. This mechanism provides
at least some kind of a generic story about why fat tails might be inherent in some situations.
The part of the distribution of possible future outcomes that we might know now (from
inductive information of a form conveyed by past data) concerns the relatively more likely
outcomes in the middle of the probability distribution. From past observations, plausible
interpolations or extrapolations, and the law of large numbers, there may be at least some
modicum of confidence in being able to construct a reasonable picture of the central regions
of the posterior-predictive PDF. As we move toward probabilities in the periphery of the
distribution, however, we are increasingly moving into the unknown territory of subjective
uncertainty, where our probability estimates of the probability distributions themselves
become increasingly diffuse because the frequencies of rare events in the tails cannot be
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pinned down by previous experiences. From past data alone, it is not possible to know enough
now about the frequencies of future extreme tail events to make the outcomes of a BCA be
independent from artificially imposed limitations on the extent of possibly catastrophic
outcomes. Climate change economics generally and the fatness of climate change tails spe-
cifically are prototypical examples of this principle, because we are trying to extrapolate
inductive knowledge far outside the range of limited past experience. To put a sharp point
on this seemingly abstract issue, the thin-tailed PDFs that implicitly support gradualist
conclusions have at least some theoretical tendency to morph into fat-tailed PDFs when
it is admitted that we are unsure about the functional forms or structural parameters be-
hind these implicitly assumed thin-tailed PDFs—at least where high temperatures are
concerned.
A fat upper tail of the PDF of lnD makes the willingness to pay (WTP) to avoid extreme
climate changes very large, indeed arbitrarily large if the coefficient of relative risk aversion is
bounded above one. In Weitzman (2009a), I presented a formal argument within a specific
mathematical structure, but this formal argument could have been embedded in alternative
mathematical structures—with the same basic message. The particular formal argument I
presented was in the form of what I called a ‘‘dismal theorem’’ (DT). In this particular for-
malization, the limiting expected stochastic discount factor is infinite (or, what I take to be
equivalent for purposes here, the limiting WTP to avoid fat-tailed disasters constitutes all of
output). Of course, in the real world, WTP is not 100 percent of output. Presumably the PDF
in the bad fat tail is thinned, or even truncated, perhaps due to considerations akin to what lies
behind the ‘‘value of a statistical life’’ (VSL)—after all, we would not pay an infinite amount to
eliminate the fat upper tail of climate change catastrophes. Alas, in whatever way the bad fat
tail is thinned or truncated, a climate change BCA based upon it might remain sensitive to the
details of the thinning or truncation mechanism because the disutility of extreme climate
change is ‘‘essentially’’ unbounded.12 Later, I discuss the meaning of this potential lack of
robustness in climate change BCA and speculate on some of its actionable consequences.
12
There is ‘‘essentially’’ unlimited liability here because global stakeholders cannot short the planet as a hedge
against catastrophic climate change.288 M. L. Weitzman
Interpreting Infinity
Disagreements abound concerning how to interpret the infinity symbol that appears in the
formulation of the DT. There is a natural tendency to scoff at economic models that yield
infinite outcomes. This reaction is presumably inspired by the idea that infinity is a ridiculous
result, and that, therefore, an economic model that yields an infinity symbol as an outcome is
fundamentally misspecified, and thus dismissable. Critics cite examples to argue earnestly
that expected disutility from climate change cannot actually be infinite, as if this were an
indictment of the entire fat-tailed methodology. I believe that, in the particular case of climate
change, the infinity symbol is trying to tell us something important. That is, the infinite limit
in the DT is a formal mathematical way of saying that structural uncertainty in the form of fat
tails is, at least in theory, capable of swamping the outcome of any BCA that disregards this
uncertainty.
The key issue here is not a mathematically illegitimate use of an infinite limit in the DT.
Infinity is a side show that has unfortunately diverted attention from the main issue, which is
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nonrobustness. It is easy to modify utility functions, to add on VSL-like restrictions, to trun-
cate probability distributions arbitrarily, or to introduce ad hoc priors that cut off or other-
wise severely dampen low values of welfare-equivalent consumption. Introducing any of these
(or many other attenuating mechanisms) formally replaces the infinity symbol by some
uncomfortably large, but finite, number. Unfortunately, removing the infinite limit in these
or other ways does not eliminate the underlying problem because it then comes back to haunt
us in the form of a WTP that is arbitrarily large in order to erase the structural uncertainty.
Just how large the WTP is can depend greatly upon obscure details about how the upper tail of
the PDF of lnD has been thinned or bounded.
As a case in point of just how fuzzy an actual upper bound might be, consider imposing
VSL-like restrictions on the WTP to avoid planet ruining climate change. Individuals make
choices all the time that involve trading off some amount of monetary wealth for some change
in the probability of exposure to fatal risks. The VSL is typically defined as the monetary
premium nM a person would be willing to pay to avoid exposure to a tiny increased prob-
ability of death nq, per increment of probability change. A large number of studies have been
used to estimate the VSL. Some time period is chosen, over which nM and nq are measured,
and then the ratio VSL ¼ nM/nq is calculated. Very rough VSL values for the United States
have typically been estimated at about $10 million.13 With average per capita income in the
United States at $50,000 per year, the VSL represents some two hundred years of income per
unit change in mortality probability. In the case of climate change, GHG concentrations of
800 ppm of CO2e imply that ProbP [T 10°C] 1/20 (see Table 2). Suppose for the sake of
argument that T 10°C represents something like the ‘‘death of humanity.’’ Then a naively
calculated per capita WTP to avoid this scenario would be about ten years’ worth of income,
a big number. The fact that this scenario would occur centuries from now, if at all, would
lower this WTP. The fact that this event represents the ‘‘death of humanity,’’ rather than the
death of a single human, would raise this WTP, perhaps considerably. Numerous other un-
derlying considerations would also affect the calculation of this WTP. My numerical example
here has many serious flaws, is easy to criticize, and is extraordinarily nonrobust. But I think it
13
See, for example, the numbers cited in Viscusi and Aldi (2003), updated into 2010 dollars.Fat-Tailed Uncertainty in the Economics of Catastrophic Climate Change 289
illustrates how difficult it can be in practice to place an upper bound on the WTP to avoid
catastrophic climate change.
Thus, although one can easily remove the infinity symbol from the DT, one cannot so easily
‘‘remove’’ the basic underlying economic problem of extreme sensitivity to fat tails and the
resulting conundrum of deciding policy under such circumstances. The overwhelming
majority of real-world BCAs have thin upper tails in lnD from limited exposure to globally
catastrophic risks. However, a few very important real-world situations have effectively
unlimited exposure due to structural uncertainty about their potentially open-ended
catastrophic reach. Climate change is unusual in potentially affecting the entire worldwide
portfolio of utility by threatening to drive all of planetary welfare to disastrously low levels in
the most extreme scenarios.
Some Implications of the DT for BCA and Policy
The ‘‘standard’’ BCA approach appears to offer a constructive ongoing scientific–economic
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research program for generating ever more precise outputs from ever more precise inputs. By
contrast, my main message can seem off-putting because it can be painted as antiscientific and
antieconomic. Fat tails and the resulting limitations they impose on the ability of BCA to
reach robust conclusions are frustrating for economists. After all, we make a living from plug-
ging rough numbers into simple models and reaching specific conclusions (more or less) on
the basis of those numbers. What quantitative advice are we supposed to provide to policy
makers and politicians about how much effort to spend on averting climate change if the
conclusions from modeling fat-tailed uncertainties are not clear cut? Practical men and
women of action have a low tolerance for vagueness and crave some kind of an answer,
so they have little patience for even a whiff of fuzziness from two-handed economists. It
is threatening for us economists to admit that constructive ‘‘can do’’ climate change BCA
may be up against some basic limitations on the ability of quantitative analysis to yield robust
policy advice. But if this is the way things are with the economics of climate change, then this
is the way things are. Nonrobustness to subjective assumptions about catastrophic outcomes
is an inconvenient truth to be lived with rather than a fact to be denied or evaded just because
it looks less scientifically objective in BCA. If this limits the ability to give fine-grained and
concrete answers to an impatient public, then so be it.
BCA is valuable, even indispensable, as a disciplined framework for organizing information
and keeping score. But all BCAs are not created equal. In rare situations with effectively
unlimited downside liability, like climate change, BCAs can be fragile to the specifications
of extreme tail events. Perhaps economists need to emphasize more openly to the policy
makers, the politicians, and the public that, while formal BCA can be helpful, in the particular
case of climate change there is a danger of possible overconfidence from undue reliance on
subjective judgments about the probabilities and welfare impacts of extreme events. What we
can do constructively as economists is to better explain both the magnitudes of the unprec-
edented structural uncertainties involved and why this feature limits what we can say, and
then present the best BCAs and the most honest sensitivity analyses that we can under fat-
tailed circumstances, including many different functional forms for extremes. At the end of
the day, policy makers must decide what to do on the basis of an admittedly sketchy economic
analysis of a gray area that just cannot be forced to render clear robust answers. The moral ofYou can also read
